Data-Driven Free-Energy Learning from Trajectories via Structure-Preserving Exponential Time Differencing

1. Introduction.

• TIFE identifies unknown free energy and optionally a positive diffusion coefficient from trajectories without energy or force labels, then drives stabilized ETD evolution preserving the learned gradient-flow structure.Under suitable assumptions, the scheme preserves the maximum-bound principle and unconditionally dissipates the original learned energy; the paper also establishes identifiability estimates and fully discrete error bounds.Source: §1 · pp. 2–3

2.1. Spatial discretization.

• The discrete maximum-bound result assumes a positivity-preserving, ℓ∞-contractive semigroup, whereas energy dissipation requires bounded upper curvature on the numerical states.The energy-dissipation theorem does not require the maximum principle.Source: §2.1 · p. 4

2.2. TIFE parameterization.

• A smooth C2 free-energy model defines the nonlinear force through its derivative, making the induced continuous model a gradient flow.The density may be represented by a smooth multilayer perceptron, spline, polynomial basis, or another C2 model.Source: §2.2 · p. 4

2.3. State-pair training objective.

• The state-pair objective identifies the TIFE density through ETD1 response factorization, using neither energy labels nor nonlinear-force labels.The response loss controls force error only conditionally because the response operator smooths and finite-step data incur a freezing defect.Source: §2.3 · p. 5, §2.4 · p. 6

2.4. From response-space training error to force error.

• Under the stated assumptions, ETD-response residuals control empirical force error, with observation error and finite-step freezing defect contributing; uniform estimates require state coverage and attention to conditioning.A small mean-square training loss alone does not ensure mesh-uniform force accuracy or control unsupported extrapolation.Source: §2.4 · pp. 6, 8
• The method retains the known Laplacian structure while inferring its scalar coefficient jointly with the local energy.Source: §2.4 · p. 8

2.5. Joint inference of a scalar Laplacian coefficient and the TIFE.

• In the joint ETD predictor, the inferred coefficient changes both the linear response and nonlinear-force propagation, rather than being fitted in a separate regression.Source: §2.5 · p. 8
• Under joint response identifiability, response error controls coefficient and force errors, and the bound extends to perturbed observations with an added noise contribution.Source: §2.5 · p. 9
• Spatially rich data with comparable states at different curvatures help distinguish the coefficient from local force; multiple initial conditions and mixed-frequency modes help.Time scale and mobility are kept fixed and known; with unknown mobility, only products involving mobility, the coefficient, and the force are generally identifiable without further normalization.Source: §2.5 · p. 10

3.1. Stabilized splitting.

• The splitting shifts the linear operator by κ and defines the stabilized nonlinearity from the TIFE density’s variational force.The shift is chosen to make the shifted operator nonnegative and its semigroup contractive.Source: §3.1 · p. 10

3.2. TIFE-based ETD1 scheme.

• The curvature-informed ETD scheme uses the TIFE-based update with a positive-definite operator factor M(τLκ,h).For an inferred Laplacian coefficient, the same update is used with the corresponding operator.Source: §3.2 · p. 11
• The operator identity is interpreted spectrally, so the update remains well-defined when L has a zero eigenvalue.The quotient is continuously extended at λ = 0, and φ1(τL) remains invertible for nonnegative L.Source: §3.2 · p. 11
• For z ≥ 0, M(z) is positive, giving the operator M(τL) positive definiteness.The scheme’s coercive reformulation relies on this positivity.Source: §3.2 · p. 11

4. Maximum-bound principle and unconditional energy stability.

• For every τ > 0, the scheme satisfies the maximum-bound principle under the stated assumptions.The proof uses a bounded stabilized nonlinearity on the admissible interval; singular potentials can be treated on a strictly separated interval when the corresponding conditions hold.Source: §4 · pp. 12–13

4.1. Unconditional dissipation of the original TIFE.

• For every τ > 0, the scheme dissipates the original TIFE functional when the states lie in the stated admissible set.“Unconditional” means no time-step restriction; both numerical states must still belong to that set.Source: §4.1 · pp. 13–14
• The maximum-bound principle supplies an invariant interval for bounding the TIFE density’s upper curvature, but energy dissipation can hold without it when another argument gives that bound.The energy proof requires states to remain in a curvature-bounded set.Source: §4.1 · p. 14
• With an inferred nonnegative Laplacian coefficient, the corresponding update retains the maximum-bound and dissipation properties under the theorems’ conditions.These properties concern the inferred energy and coefficient pair; the coefficient need not be close to its true value.Source: §4.1 · p. 14

5.1. Assumptions for the error estimate.

• The error bounds assume a common compact admissible interval, bounded force regularity, spatial consistency, and stable state restriction.The stated regularity example is u ∈ L∞(0,T;H4(Ω)) and ut ∈ L∞(0,T;L2(Ω)); nodal restriction needs an additional stability condition.Source: §5.1 · p. 15

5.2. Nonlinear perturbation estimate.

• The nonlinear perturbation estimate separates state error from force-inference error: |Hθ(v) − H(w)| ≤ Lθ|v − w| + δF.The grid estimate adds the corresponding discrete norm term and the interval-size factor ∥1∥hδF.Source: §5.2 · p. 15

5.3. Local consistency of the TIFE-based ETD1 scheme.

• The ETD1 local consistency error is O(τ² + τh²) under the stated regularity and spatial-consistency assumptions.The two contributions arise from time variation and spatial consistency, respectively.Source: §5.3 · p. 16

5.4. Fully discrete error estimate.

• For 0 < τ ≤ τ0 and 0 < h ≤ h0, the fully discrete error is bounded by CT(τ + h² + δF), with no coupling condition between τ and h.The estimate separates temporal and spatial discretization from TIFE-induced force-inference error.Source: §5 · p. 14, §5.4 · pp. 17–18
• The training-residual rollout bound adds observation perturbations, ETD1 freezing defect, response-inversion conditioning, and state-space coverage to deployment discretization errors.The numerical tables assess fixed-coefficient potential identification, not synthetic recovery of the jointly inferred coefficient.Source: §5.4 · p. 18, §5.5 · p. 19

5.5. Additional error from the inferred Laplacian coefficient.

• With an inferred Laplacian coefficient, the residual bound includes coefficient error through an O(h² + δa) consistency term.Closing the joint-training-to-rollout chain is conditional on joint identifiability and the rollout remaining within the data-supported interval.Source: §5.5 · pp. 18–19

6.8. MBP and unmodified learned-energy dissipation.

• The endpoint-constrained scheme gives the maximum-bound principle and dissipation of the original learned energy; at T=4, maximum magnitude remains below β.The measured maximum magnitude is 0.993598 < β, while the largest measured energy increase and inequality residual remain below the stated float32 verification tolerance.Source: §6.2 · p. 21, §6.8 · p. 25, Figure 6.6 · p. 26

6.4. Recovery of the potential and nonlinear force.

• Curvature recovery remains approximately 1.25% error despite being the most sensitive diagnostic because it differentiates the learned network twice.Across 12 unseen initial conditions at T=4, mean rollout error is 1.1974 × 10^-3 (standard deviation 4.4248 × 10^-4).Source: §6.3 · p. 21, Table 6.2 · p. 22, Table 6.3 · p. 22

6.5. Unseen initial conditions and long-time prediction.

• Below 0.8% relative error at T=50, the frozen learned model extrapolates from training only through T=1.2 without spurious oscillations or incorrect phase topology.The phase-field snapshots compare learned-potential ETD1 with a fine-step reference.Source: §6.5 · p. 22, Figure 6.3 · p. 23, Table 6.4 · p. 23

6.6. Comparison with matched-size learned representations.

• At matched parameter counts, potential learning reduces direct-force mean error by approximately 59% and is more accurate and robust than the residual FNO in this experiment.The comparison is specific to the tested data, architectures, and parameter budget.Source: Table 6.5 · p. 24, §6.6 · p. 24, §6.5 · p. 24

6.7. Transfer without retraining.

• With weights frozen, the method transfers across grids N=32, 48, 64, and 96, ε=0.05–0.2, and time steps up to 0.08, four times the training step.Errors remain nearly unchanged across the tested grids; error increases as the deployment step grows.Source: §6.7 · pp. 24–25, Table 6.6 · p. 25, Figure 6.5 · p. 25

6.9. Five-seed studies of data, noise, and time steps.

• More than one order of magnitude: increasing state pairs from 160 to 640 reduces mean rollout error, while force error falls from approximately 7.6% to 0.67%.At 0.1% relative observation noise, rollout error changes only slightly; tested rollouts remain stable and retain the MBP and learned-energy monotonicity.Source: Table 6.8 · p. 27, §6.8 · p. 27, Table 6.9 · p. 27
• At matched budgets of 640 versus 639 pairs, pooled time-step training is less deployment-step-sensitive but has larger absolute error.The authors attribute the trade-off to first-order modified-equation biases in ETD1 observations at different time steps.Source: §6.9 · pp. 27–28, Table 6.10 · p. 28, Figure 6.9 · p. 29

6.10. Non-polynomial Flory–Huggins potential.

• At T=20, the Flory–Huggins rollout error is approximately 1.2%, stays inside [−0.9, 0.9], and reproduces the reference coarsening pattern.This test uses hard inward endpoint-slope signs, a stronger problem-specific separation prior.Source: Table 6.11 · p. 29, §6.10 · p. 29, Figure 6.11 · p. 30

1. Low-dimensional mechanism identification.

• The method learns a scalar potential instead of a high-dimensional space–time solution or full state-update operator, yielding accurate force recovery from short trajectories and an interpretable energy.Source: §1 · p. 31

2. Separation of known and unknown physics.

• The ETD solver retains the known stiff diffusion operator, while the neural network models only the missing constitutive term.This avoids spending model capacity on relearning known spatial dynamics.Source: §2 · p. 31

3. Long-time and cross-configuration reuse.

• A potential trained up to T = 1.2 remains accurate and stable as far as T = 50.Source: §3 · p. 31
• The same weights transfer to unseen initial conditions, grids from 322 to 962, diffusion parameters from 0.05 to 0.2, and time steps up to four times the training step.Source: §3 · p. 31

4. Structure preservation without an auxiliary energy.

• The curvature-informed deployments satisfy the tested MBP and original learned-energy dissipation diagnostics.Source: §4 · p. 31
• The deployments introduce no SAV/GSAV variable or modified auxiliary energy.Source: §4 · p. 31

• Force and rollout errors consistently decrease as the number of state pairs increases.Source: §5 · p. 31
• The method is insensitive to 0.1% relative noise.Source: §5 · p. 31
• At higher noise levels, performance degrades gradually.Source: §5 · p. 31

• The potential representation is more accurate than a parameter-matched direct-force network and substantially more robust than a residual FNO baseline.Source: §6 · p. 31
• The compelling regime is a known differential operator, unknown local potential, short-time observations, and many long-time or multi-initial-condition predictions.Source: §6 · p. 31
• The experiments do not establish superiority over every PINN or neural-operator variant.Source: §6 · p. 31
• They also do not establish a computational advantage for a single forward solve when the complete potential is already known.Matched wall-clock, memory, and accuracy comparisons are left for future work.Source: §6 · p. 31

7. Conclusions and extensions

• Under stated semigroup, endpoint, and upper-curvature assumptions, the TIFE-based ETD1 map preserves the admissible maximum bound and unconditionally dissipates the original discrete TIFE functional.The method uses inferred curvature for stabilization without an SAV/GSAV auxiliary variable or modified energy.Source: §7 · p. 31
• The error bounds are O(τ + h^2 + δF) for a fixed operator and O(τ + h^2 + δF + δa) when the coefficient is inferred.The response-to-force estimate links δF to response residual, observation perturbations, finite-step freezing defect, and state fill distance; joint identifiability also lets the coefficient-dependent response residual control δa and empirical force error.Source: §7 · p. 31
• State-only training recovers quartic and logarithmic potentials, supports accurate long-time rollouts beyond the observation window, and transfers across initial conditions, grids, diffusion parameters, and time steps without retraining.Curvature-informed deployments satisfy the tested maximum-bound and learned-energy diagnostics; five-seed studies assess data scaling and observation-noise robustness.Source: §7 · p. 31
• The reported numerical tables use a supplied Laplacian coefficient, so they do not empirically validate joint coefficient recovery.Source: §7 · p. 32
• Proposed extensions include higher-order ETD schemes, certified curvature bounds, vector-valued order parameters, multi-parameter elliptic operators, and nonlocal or mobility-dependent energies.Source: §7 · p. 32